Revisiting Korovkin-type Theorems in Banach Function Spaces

TL;DR

This paper revisits Korovkin-type theorems in Banach function spaces, showing positivity isn't always necessary, providing new operator versions, quantitative forms, and practical examples including Lebesgue and weighted spaces.

Contribution

It demonstrates that positivity of operators is not essential in Korovkin-type theorems within Banach function spaces and introduces new operator and quantitative versions.

Findings

Positivity of operators is not necessary for Korovkin-type theorems.

Provides an operator version of the theorem under positivity assumptions.

Includes numerical illustrations and applications to various function spaces.

Abstract

This article delves into Korovkin-type theorems in Banach function spaces, as established by Yusuf Zeren et al. (2022). We prove that in this theorem, the positivity of the operators is not a necessary requirement and provide example of a non positive operator where it is applicable. Under the assumption of positivity, we establish an operator version of the result. Additionally, we derive a quantitative form of the result using the modulus of continuity. We apply the result to examples such as Lebesgue space, Weighted Lebesgue space, Grand Lebesgue space, etc. Furthermore, we present numerical illustrations for specific cases.